A discrete-time homing problem with two optimizers

A stochastic difference game is considered in which a player wants to minimize the time spent by a controlled one-dimensional symmetric random walk {𝑋𝑛,𝑛=0,1,…} in the continuation region 𝐶:={1,2,…}, and the second player seeks to maximize the survival time in C. The process starts at 𝑋0=𝑥>0 and the game ends the first time 𝑋𝑛≤0. An exact expression is derived for the value function, from which the optimal solution is obtained, and particular problems are solved explicitly.